Surveying - The Science of Navigation - page 85

Because of the uncertainty of the angles due to measurement error, there is
an uncertainty about the forest fire location, shown as the shaded area
surrounding C . Note that the size of the ''uncertainty box'' varies with
position: another fire at D has a di√erent-shaped box, even though D is the
same distance, d , from the baseline. (The error is a minimum when the
angle ADB is 90\.) A fire farther away at E is surrounded by a much larger
''uncertainty box.'' This example shows how the errors of triangulation
depend upon the shape and size of the triangles.
Surveying by triangulation is a very old practice. It may have existed in
Han China in the third century BCE. It certainly was used in Arab lands
following the Islamic expansion in the seventh century CE, and from the
Arabs it came to be known in Christian Spain before the Reconquest. From
Spain, however, the knowledge di√used to the rest of Europe only slowly.
The idea was proposed as a method for mapping by the Dutch cartographer
Gemma Frisius in 1533; eighty years later the modern systematic form
of triangulation was worked out and practiced by the Dutchman Wille-
brord Snell, whom we met earlier. He showed how, in large-scale geod-
etic surveys, the curvature of the earth could be taken into account. Mod-
ern triangulation can turn meshes of triangles into three-dimensional
maps displayed on computer screens, showing topography projected onto a
screen like a photo. For such cases there is a best choice of triangle vertices
(i.e., of survey points). Triangulation with such a set of points, with a view
to converting the data points into a 3-D image, is known as Delaunay
triangulation . 17
TRILATERATION
Whereas triangulation estimates distances via angles, trilateration does the
reverse, estimating the angles or positions of a point given certain dis-
tances. It is the method that has to be adopted by GPS receivers, since GPS
provides detailed distance information and no explicit angle information. I
will outline trilateration here in the context of GPS position estimation.
The math of trilateration is complicated, but the basic idea is geometrical
and quite easy to convey.
Consider first a two-dimensional trilateration problem. In figure 3.21a
we have three satellites at locations A , B , and C . They transmit data to a re-
ceiver, and from this information the receiver software is able to calculate
17. Most surveying textbooks include overviews and detailed analyses of triangulation.
See, e.g., Davis (1998), Duggal (2004), Kavanagh (2008), Liu (1997), Petrie and Kennie
(1987), and Pugh (1975).